#455641

order	512
nilpotency class	3
isoclinism	isoclinic class of #455641
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

927 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,9,13,11,16,10,15)(7,17,18,26,21,28,20,12)(8,22,23,29,24,14,25,30)(19,31,33,27,34,36,32,35);(1,2)(3,4)(5,7,10,18,11,21,9,20)(6,12,15,17,16,26,13,28)(8,19,23,32,24,34,25,33)(14,27,30,31,22,35,29,36);(2,4)(6,13)(7,18)(8,23)(9,10)(12,26)(15,16)(20,21)(24,25)(27,31)(29,30)(32,33)(35,36);(5,8)(6,14)(7,19)(9,25)(10,23)(11,24)(12,27)(13,29)(15,30)(16,22)(17,31)(18,32)(20,33)(21,34)(26,35)(28,36)

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362837);